Salicylic acid peels are treatments used to deeply exfoliate the surface layers of the skin at a pore level. This BHA breaks the bonds that hold the skin's cells together, causing them to shed away or peel depending on the concentration used.  
Skin is rejuvenated after skin regeneration process is complete.
Compared to other types of chemical peels, salicylic acid is a superficial Peel that provides a gentle exfoliating action Which makes it safer for use at home.
For advance level, stronger  salicylic acid peels at higher concentration which  penetrates more deeply are usually administered by professionals.
Once the preferred concentration has been purchased and ready for use, you want to make sure that your skin is thoroughly cleansed. 
The skin is degreased with alcohol or an astringent that gets rid of excess oil on the skin and petroleum jelly is applied on sensitive areas like nostrils, etc.
After, the salicylic acid solution is applied to the skin with cotton pads or applicator brush. Total application depends how strong the peel should be, the application is done in layers where after one application of the solution is layered on top of the other within few seconds to mins. 
The solution is left on your skin for about three to five minutes. Some individuals may feel some mild burning or stinging while some feel nothing. 
After the time is up, the peel is neutralized with water or a neutralizing solution. To make a neutralizing solution, make a 20% sodium bicarbonate solution with water. 
 After neutralizing the acid, a mild cleanser may be used to wash off the left over solution and then a  moisturizer is applied.
Note that Salicylic acid is a superficial peel, meaning not everyone will experience peeling from salicylic acid but it is sure to soften, brighten the face and visibly reduce excess oil productions which result to decrease in acne formaction. 
For more severe skin issues, series of peels session is needed mostly once or twice in a week application depending on your desired result.
If using Salicylic acid peel for tough areas like hands, knuckles , feet, etc, you will need to apply the salicylic acid for 3 consecutive days  so enough amount of the peel is absorbed by the skin to cause peeling.